What is the ruling on a woman who has forbidden herself from marrying a specific young man until the Day of Judgment if she performs a certain act, and then she performs that act? What is the expiation for that, and is it permissible for her to marry this young man?

Prohibiting the lawful does not prohibit anything except one's wife. If a man prohibits something other than his wife, the predominant opinion is that expiation becomes obligatory, because Allah Almighty referred to prohibition as an oath in His saying: "O Prophet, why do you prohibit what Allah has made lawful for you, seeking to please your wives? And Allah is Forgiving and Merciful. Allah has already ordained for you the dissolution of your oaths." So, whoever prohibits something lawful other than his wife, or conditions the prohibition of something lawful on a condition, that thing does not become prohibited, and he must offer an expiation for an oath if he fulfills the condition upon which the prohibition was made dependent.